×

zbMATH — the first resource for mathematics

A sensitivity analysis for linear systems involving M-matrices and its application to the Leontief model. (English) Zbl 0556.15003
Let \(M^ 0\) and \(M^*\) be (n\(\times n)\) non-singular M-matrices (n\(\geq 2)\). (That is: \(M^ 0\) and \(M^*\) may be written as \(M^ 0=\lambda I- A^ 0\), \(M^*=\lambda I-A^*\) where \(\lambda\) is a positive scalar, and \(A^ 0\) and \(A^*\) are non-negative matrices, and \((M^ 0)^{- 1}\), \((M^*)^{-1}\) are also non-negative.) \(M^ 0\) and \(M^*\) may differ only in the first s \((0<s<n)\) columns. Let \(w^ 0\) and \(w^*\) be strictly positive n-vectors, which coincide in their last (n-s) entries. Let \(p^ 0=(M^ 0)^{-1}w^ 0\), \(p^*=(M^*)^{-1}w^*\). Then if \((p^ 0M^*)_ i>w^*_ i\) for \(i\in S\), the authors prove \(\min_{i\in S}\{p^*_ i/p^ 0_ i\}<\min_{i\in R}\{p^*_ i/p^ 0_ i\}\), where \(S=\{1,2,...,s\}\), \(R=\{s+1,...,n\}\). This is a partial generalization of Theorem 21 of G. Sierksma [Linear Algebra Appl. 26, 175-201 (1979; Zbl 0409.90027)]; see also the reviewer’s paper [Non-negative matrices and Markov chains (1981; Zbl 0471.60001 pp. 35- 39], in that changes in the M-matrix \(\{\) from \(M^ 0\) to \(M^*\}\) are also permitted.

MSC:
15A06 Linear equations (linear algebraic aspects)
93B35 Sensitivity (robustness)
15B48 Positive matrices and their generalizations; cones of matrices
91B60 Trade models
PDF BibTeX XML Cite
Full Text: DOI
References:
[1] Berman, A.; Plemmons, R.J., Nonnegative matrices in the mathematical sciences, (1979), Academic New York · Zbl 0484.15016
[2] Debreu, G.; Herstein, I.N., Nonnegative square matrices, Econometrica, 21, 567-607, (1953) · Zbl 0051.00901
[3] Elsner, L.; Johnson, C.R.; Neumann, M., On the effect of the perturbation of a nonnegative matrix on its Perron eigenvector, Czechoslovak math. J., 32, 99-109, (1982) · Zbl 0501.15010
[4] Fiedler, M.; Pták, V., On matrices with non-positive off-diagonal elements and positive principal minors, Czechoslovak math. J., 12, 382-400, (1962) · Zbl 0131.24806
[5] Fujimoto, T., An elementary proof of Okishio’s theorem for models with fixed capital and heterogeneous labour, Metroeconomica, 33, 21-27, (1981)
[6] T. Fujimoto, C. Herrero, and A. Villar, Technical changes and their effects on the price structure, Metroeconomica, to appear.
[7] Metzler, L.A., A multiple-country theory of incomes transfers, J. political economy, 59, 14-29, (1951)
[8] Metzler, L.A., Taxes and subsidies in Leontief’s input-output model, Quart. J. econom., 65, 433-438, (1951)
[9] Morishima, M., Equilibrium, stability and growth, (1964), Clarendon Oxford · Zbl 0117.15406
[10] Murata, Y., Mathematics for stability and optimization of economic systems, (1977), Academic New York
[11] Seneta, E., Nonnegative matrices, (1973), Wiley New York · Zbl 0278.15011
[12] Seneta, E., Nonnegative matrices and Markov chains, (1981), Springer Berlin · Zbl 0471.60001
[13] Sierksma, G., Non-negative matrices: the open Leontief model, Linear algebra appl., 26, 175-201, (1979) · Zbl 0409.90027
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.